We prove two integral transformations that relate different constructions of rational approximations to . The first one relates a double integral over the unit square and a Barnes-type integral. The second one relates two Barnes-type integrals and was discovered and proved by W. Zudilin using an automated proof method. Here we offer a proof based on more classical means such as contiguous relations, the second Barnes lemma and the duplication formula for the gamma function.
"Two integral transformations related to $\zeta(2)$." Mosc. J. Comb. Number Theory 9 (4) 441 - 452, 2020. https://doi.org/10.2140/moscow.2020.9.441